Selmer groups and central values of L-functions for modular forms
Annales de l'Institut Fourier, Volume 67 (2017) no. 3, p. 1231-1276

In this article, we construct an Euler system using CM cycles on Kuga–Sato varieties over Shimura curves and show a relation with the central values of Rankin–Selberg L-functions for elliptic modular forms and ring class characters of an imaginary quadratic field. As an application, we prove that the non-vanishing of the central values of Rankin–Selberg L-functions implies the finiteness of Selmer groups associated to the corresponding Galois representation of modular forms under some assumptions.

Dans cet article, nous construisons un système d’Euler en utilisant les cycles CM sur les variétés de Kuga–Sato au-dessus de courbes de Shimura, et montrons une relation avec les valeurs centrales de fonctions L de Rankin–Selberg associées aux formes modulaires de poids 2 et aux caractères de classes d’un corps quadratique imaginaire. Comme application, nous prouvons que la non-annulation des valeurs centrales de fonctions L de Rankin–Selberg implique la finitude des groupes de Selmer associés à la représentation galoisienne de la forme modulaire sous certaines hypothèses.

Received : 2016-01-16
Revised : 2016-09-15
Accepted : 2016-09-16
Published online : 2017-05-31
DOI : https://doi.org/10.5802/aif.3108
Classification:  11F67,  11R23
Keywords: Modular forms, Selmer groups, Bloch–Kato conjecture
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     author = {Chida, Masataka},
     title = {Selmer groups and central values of $L$-functions for modular forms},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {67},
     number = {3},
     year = {2017},
     pages = {1231-1276},
     doi = {10.5802/aif.3108},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2017__67_3_1231_0}
}
Chida, Masataka. Selmer groups and central values of $L$-functions for modular forms. Annales de l'Institut Fourier, Volume 67 (2017) no. 3, pp. 1231-1276. doi : 10.5802/aif.3108. http://www.numdam.org/item/AIF_2017__67_3_1231_0/

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